The main result of the current section is Proposition REF , which shows that the constrained least squares and the ridge estimators match the lower bound (REF ) under \(L_{4}\) –\(L_{2}\)  moment equivalence assumptions specified below. We remark, however, that tighter lower bounds than that of (REF ) might be possible under some of the assumptions considered below. In particular, it is well-known that assumptions closely related to moment equivalence considered below (e.g., small-ball [1]}) might simplify the quadratic process, which gives rise to the \(r^{2}d^{2}\log (\min \lbrace n, d\rbrace )/n\)  term in Theorem REF . However, as discussed above, the multiplier term is responsible for the suboptimality of constrained least squares and in this section we are mainly trying to understand the conditions sufficient to ensure a well-behaved multiplier term.
